Quantum jumps of a qubit are usually observed between its energy eigenstates (eigenstates of sigma_z)), also known as its longitudinal component. Is it possible, instead, to observe quantum jumps between the transverse superpositions (sigma_x) of these eigenstates ?By combining a superconducting qubit-cavity system with two pump tones, we demonstrate that we can synthesize a Hamiltonian in which the cavity is dispersively coupled to the sigma_x operator of the qubit. Thus, we can perform a quantum non-demolition measurement of sigma_x and observe quantum jumps between its eigenstates. This protocol is a simple example of the variety of artificial quantum operations that can be created by driven-dissipative engineering. Such operations are at the heart of quantum simulation, and necessary for quantum error correction.